Subgroups — #28506437

Order 15104.

312 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425861	C₂	1	1	yes	maximal, characteristic, complemented
2	#21425725	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837185	C₂	1	1	yes	maximal, characteristic
4	#21423897	C₂	1	1	yes	maximal, characteristic
5	#21422184	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422182	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273733	C₂²	1	1	yes	characteristic
9	C₂×#14273667	C₂²	1	1	yes	characteristic
10	C₂×#14273667	C₂²	1	1	yes	characteristic
11	#16837384	C₄	1	1	yes	characteristic
12	#16837375	C₄	1	1	yes	characteristic
13	#16837357	C₂²	1	1	yes	characteristic
14	#16837222	—	2	2	no	—
15	#16837187	—	2	2	no	—
16	#16837185	—	2	1	no	—
17	#16837185	—	2	1	no	—
18	#16837174	C₄	1	1	yes	characteristic
19	#16837172	C₄	1	1	yes	characteristic
20	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
22	C₅₉×#405	C₂²	1	1	yes	characteristic
23	C₅₉×#346	—	2	1	no	—
24	C₄×C₁₆×C₅₉	—	2	1	no	—
25	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
26	#14273733	—	2	1	no	—
27	#14273733	—	2	1	no	—
28	#14273733	—	2	1	no	—
29	#14273733	—	2	1	no	—
30	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
31	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
32	C₂×#11745659	—	2	1	no	—
33	C₂×C₈×D₅₉	—	2	1	no	—
34	#14273686	C₂×C₄	1	1	yes	characteristic
35	#14273667	—	2	1	no	—
36	#14273667	—	2	1	no	—
37	#14273667	—	2	1	no	—
38	#14273667	—	2	1	no	—
39	C₂×C₅₉⋊C₁₆	—	2	1	no	—
40	C₂×C₅₉⋊C₁₆	—	2	1	no	—
41	#14273645	C₈	1	2	yes	—
42	#14273643	C₈	1	2	yes	—
43	#14273641	—	2	2	no	—
44	#14273637	—	2	1	no	—
45	C₈×Dic₅₉	—	2	1	no	—
46	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
48	#117×C₅₉	C₂×C₄	1	1	yes	characteristic
49	M₃₂×C₅₉	—	2	1	no	—
50	M₃₂×C₅₉	D₄	1	2	yes	—
51	C₂×C₁₆×C₅₉	—	2	1	no	—
52	C₂×C₁₆×C₅₉	—	2	1	no	—
53	C₂×C₁₆×C₅₉	—	2	1	no	—
54	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
55	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
56	#97×C₅₉	—	2	1	no	—
57	C₄×C₈×C₅₉	—	2	1	no	—
58	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₂²⋊D₅₉	—	2	1	no	—
60	#11745685	—	2	1	no	—
61	#11745685	—	2	1	no	—
62	#11745685	—	2	1	no	—
63	#11745685	—	2	1	no	—
64	C₂×D₂₃₆	C₂×C₈	1	1	yes	characteristic
65	C₂×C₄×D₅₉	—	2	1	no	—
66	C₂×Dic₁₁₈	C₂×C₈	1	1	yes	characteristic
67	M₁₆×C₅₉	—	2	1	no	—
68	M₁₆×C₅₉	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₅₉	—	2	1	no	—
70	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
73	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
74	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
75	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
76	C₄⋊C₄×C₅₉	—	2	1	no	—
77	C₂²⋊C₄×C₅₉	M₁₆	1	2	yes	—
78	C₄²×C₅₉	—	2	1	no	—
79	C₂²⋊Dic₅₉	—	2	1	no	—
80	C₄⋊Dic₅₉	—	2	1	no	—
81	C₄×Dic₅₉	—	2	1	no	—
82	#11745664	—	2	1	no	—
83	#11745664	—	2	1	no	—
84	#11745664	—	2	1	no	—
85	#11745664	C₂×C₈	1	2	yes	—
86	C₂×C₅₉⋊C₈	—	2	1	no	—
87	C₂×C₅₉⋊C₈	—	2	1	no	—
88	#11745659	—	4	1	no	—
89	#11745659	—	4	1	no	—
90	C₈×D₅₉	—	4	1	no	—
91	C₈×D₅₉	—	4	1	no	—
92	C₁₆×C₅₉	—	2	2	no	—
93	C₁₆×C₅₉	—	2	1	no	—
94	C₁₆×C₅₉	—	2	1	no	—
95	C₅₉⋊C₁₆	—	4	1	no	—
96	C₅₉⋊C₁₆	—	4	1	no	—
97	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
98	C₂²×D₅₉	—	2	1	no	—
99	C₂×C₄×C₅₉	—	2	2	no	—
100	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
105	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
106	C₂²⋊D₅₉	—	4	1	no	—
107	C₂²⋊D₅₉	—	4	1	no	—
108	C₂×Dic₅₉	—	2	1	no	—
109	C₂×Dic₅₉	—	4	1	no	—
110	D₂₃₆	—	2	1	no	—
111	D₂₃₆	—	2	1	no	—
112	C₄×D₅₉	—	4	1	no	—
113	C₄×D₅₉	—	4	1	no	—
114	Dic₁₁₈	—	2	1	no	—
115	Dic₁₁₈	—	2	1	no	—
116	C₈×C₅₉	—	2	2	no	—
117	C₈×C₅₉	#121	1	1	yes	characteristic
118	C₈×C₅₉	#123	1	1	yes	characteristic
119	C₈×C₅₉	#124	1	1	yes	characteristic
120	C₈×C₅₉	#126	1	1	yes	characteristic
121	C₅₉⋊C₈	—	2	2	no	—
122	C₅₉⋊C₈	—	4	1	no	—
123	#13123	—	59	1	no	maximal, Sylow
124	C₂²×C₅₉	—	4	1	no	—
125	C₂²×C₅₉	#431	1	1	yes	characteristic
126	C₂²×C₅₉	#435	1	1	yes	characteristic
127	C₂²×C₅₉	#436	1	1	yes	characteristic
128	C₂×D₅₉	—	4	1	no	—
129	C₂×D₅₉	—	8	1	no	—
130	C₄×C₅₉	—	4	2	no	—
131	C₄×C₅₉	#388	1	1	yes	characteristic
132	C₄×C₅₉	#433	1	1	yes	characteristic
133	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
134	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
135	Dic₅₉	—	4	1	no	—
136	Dic₅₉	—	8	1	no	—
137	#2153	—	59	1	no	—
138	#2117	—	59	1	no	—
139	C₂×#350	—	59	1	no	—
140	#2108	D₅₉	1	1	yes	characteristic, complemented
141	#1757	—	59	1	no	—
142	#1379	—	59	1	no	—
143	#1378	—	59	1	no	—
144	C₂×C₅₉	—	2	1	no	—
145	C₂×C₅₉	—	2	1	no	—
146	C₂×C₅₉	—	2	1	no	—
147	C₂×C₅₉	#1923	1	1	yes	characteristic
148	D₅₉	—	8	1	no	—
149	C₂×#131	—	59	1	no	—
150	C₂×M₃₂	—	59	1	no	—
151	C₂×M₃₂	—	59	1	no	—
152	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
153	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
154	#413	—	59	1	no	—
155	#410	—	59	1	no	—
156	#405	—	59	1	no	—
157	#405	C₂×D₅₉	1	1	yes	characteristic
158	#364	—	118	2	no	—
159	#350	—	118	2	no	—
160	#350	—	118	1	no	—
161	#350	—	118	1	no	—
162	#346	—	2	1	no	—
163	C₄×C₁₆	—	2	1	no	—
164	#344	—	59	1	no	—
165	#344	—	59	1	no	—
166	C₅₉	#13123	1	1	yes	characteristic, complemented, Sylow
167	C₂×#41	—	59	1	no	—
168	#131	—	118	1	no	—
169	#131	—	118	1	no	—
170	#131	—	118	1	no	—
171	#131	—	118	1	no	—
172	C₂×M₁₆	—	59	1	no	—
173	C₂×M₁₆	—	59	1	no	—
174	C₂×M₁₆	—	118	1	no	—
175	C₂×M₁₆	C₄×D₅₉	1	1	yes	characteristic
176	C₂²×C₈	—	118	1	no	—
177	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
178	#117	—	59	1	no	—
179	#117	C₄×D₅₉	1	1	yes	characteristic
180	M₃₂	—	2	1	no	—
181	M₃₂	—	118	1	no	—
182	M₃₂	—	118	1	no	—
183	M₃₂	—	118	1	no	—
184	M₃₂	—	118	1	no	—
185	M₃₂	D₂₃₆	1	2	yes	—
186	C₂×C₁₆	—	2	1	no	—
187	C₂×C₁₆	—	2	1	no	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	—	118	1	no	—
190	C₂×C₁₆	—	118	1	no	—
191	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
192	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
193	#101	—	59	2	no	—
194	#100	—	59	2	no	—
195	#99	—	118	2	no	—
196	#97	—	2	1	no	—
197	#97	—	118	1	no	—
198	C₄×C₈	—	2	1	no	—
199	C₄×C₈	—	118	1	no	—
200	#41	—	118	1	no	—
201	#41	—	118	1	no	—
202	#41	—	118	1	no	—
203	#41	—	118	1	no	—
204	C₂×Q₈	—	59	1	no	—
205	C₂×D₄	—	59	1	no	—
206	C₂×D₄	—	118	1	no	—
207	C₂²×C₄	—	118	1	no	—
208	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
209	M₁₆	—	2	1	no	—
210	M₁₆	—	59	2	no	—
211	M₁₆	—	118	1	no	—
212	M₁₆	—	118	1	no	—
213	M₁₆	—	118	1	no	—
214	M₁₆	—	236	1	no	—
215	M₁₆	—	236	1	no	—
216	M₁₆	C₈×D₅₉	1	2	yes	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	118	1	no	—
219	C₂×C₈	—	118	1	no	—
220	C₂×C₈	—	236	1	no	—
221	C₂×C₈	—	236	1	no	—
222	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
223	C₂×C₈	#11745685	1	1	yes	characteristic
224	C₂×C₈	#11745685	1	1	yes	characteristic
225	C₂×C₈	#11745685	1	1	yes	characteristic
226	C₂×C₈	#11745685	1	1	yes	characteristic
227	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	118	1	no	—
230	C₂²⋊C₄	—	118	1	no	—
231	C₂²⋊C₄	#11745659	1	2	yes	—
232	C₄²	—	2	1	no	—
233	C₄²	—	118	1	no	—
234	C₁₆	—	2	2	no	—
235	C₁₆	—	2	1	no	—
236	C₁₆	—	2	1	no	—
237	C₁₆	—	236	1	no	—
238	C₁₆	—	236	1	no	—
239	C₂³	—	118	1	no	—
240	C₂³	C₂×#11745659	1	1	yes	characteristic
241	Q₈	—	118	1	no	—
242	Q₈	—	118	1	no	—
243	D₄	—	118	1	no	—
244	D₄	—	118	1	no	—
245	D₄	—	236	1	no	—
246	D₄	—	236	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	118	1	no	—
249	C₂×C₄	—	236	1	no	—
250	C₂×C₄	—	236	1	no	—
251	C₂×C₄	—	236	1	no	—
252	C₂×C₄	#14273684	1	1	yes	characteristic
253	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
254	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
255	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
256	C₂×C₄	#14273733	1	1	yes	characteristic
257	C₂×C₄	#14273733	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	118	2	no	—
260	C₈	—	236	1	no	—
261	C₈	#14273701	1	1	yes	characteristic
262	C₈	#14273710	1	1	yes	characteristic
263	C₈	#14273715	1	1	yes	characteristic
264	C₈	#14273721	1	1	yes	characteristic
265	C₂²	—	4	1	no	—
266	C₂²	—	472	1	no	—
267	C₂²	—	236	1	no	—
268	C₂²	#16837453	1	1	yes	characteristic
269	C₂²	#16837473	1	1	yes	characteristic
270	C₂²	#16837481	1	1	yes	characteristic
271	C₄	—	4	2	no	—
272	C₄	—	472	1	no	—
273	C₄	—	236	1	no	—
274	C₄	#16837293	1	1	yes	characteristic
275	C₄	#16837461	1	1	yes	characteristic
276	C₄	C₈×C₂²⋊D₅₉	1	1	yes	characteristic
277	C₄	C₈×D₂₃₆	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	472	1	no	—
282	C₂	#21424702	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14273701
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#2108	C₂
socle	C₂×C₅₉	#1923
prime radical	C₂×#11745685	C₂×C₄
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#2108	D₅₉
hypocenter	C₅₉	#13123
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2108

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #21425861	C₂	semidirect
5. #21422184	C₂	semidirect
6. #21422182	C₂	semidirect
7. C₅₉×#2108	C₂	semidirect
140. #2108	D₅₉	semidirect
166. C₅₉	#13123	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21425861	2
#21425725	2
C₂×#16837185	2
#21423897	2
#21422184	2
#21422182	2
C₅₉×#2108	2
#13123	59